To solve this saddle point problem, we follow the operator preconditioning
framework of Hiptmair and Mardal & Winther [1]}, [2]},
employing a block-diagonal preconditioner with the Riesz maps (\(\alpha =\beta =1\) )
for \(V^{k-1}\)  and \(V^k\) . We use 4 Chebyshev iterations preconditioned by
the SC-PH solver for each block. For \(L^2\) , we use 4 Chebyshev iterations
preconditioned by point-Jacobi in the FDM
basis for \(\mathrm {DQ}_{p-1}\) . As outer Krylov solver we
employ the minimum residual method (MINRES) [3]}, as this allows for
the solution of indefinite problems using a symmetric coercive preconditioner.
The convergence criterion for the iteration is a relative reduction of the
(natural) \(P^{-1}\) -norm of the residual by a factor of \(10^8\) , starting from a
zero initial guess. The iteration counts for the cases \(k=1,2,3\) 
are reported in tab:hodge-iter. As for the Riesz maps, we observe robustness with respect
to both \(h\)  and \(p\) .
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